Problem: Determine how many solutions exist for the system of equations. ${-8x-2y = 2}$ ${y = -4x-3}$
Solution: Convert both equations to slope-intercept form: ${-8x-2y = 2}$ $-8x{+8x} - 2y = 2{+8x}$ $-2y = 2+8x$ $y = -1-4x$ ${y = -4x-1}$ ${y = -4x-3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x-1}$ ${y = -4x-3}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.